As the DM, the tools of your trade are dice — platonic solid-shaped or just about any other sort. The random numbers you generate by rblling dice determine the results based on the probabilities determined herein or those you have set forth on your own. In case you are not familiar with probability curves, there are two types which are determined by your dice: linear (straight line),'which has equal probability of any given integer in the number group, and bell (ascending and descending line), which has greater probability towards the center of the group of numbers than at either end. The two curves are illustrated thus:
[Image of linear curve for 10-sided dice, with each number having a 10% chance of coming up.]
Linear probability develops a straight line of ascending probability when used as a cumulative probability as shown above.
Bell distribution, when used to delineate the probability of certain numbers appearing, develops o curved line like this:
[Image of a bell curve for a 3d6 roll, with 3 and 18 being the least probable and 8 through 12 being the most probable.]
A single die, or multiple.dice read in succession (such as three dice read as hundreds, tens and decimals) give linear probabilities. Two or more dice added together generate a bell-shaped probability curve.
Before any further discussion takes place, let us define the accepted abbreviations for the various dice. A die is symbolized by "d", and its number of sides is shown immediately thereafter. A six-sided die is therefore "d6", d8 is an eight-sided die, and so on.-Two four-sided dice are pressed by 2d4, five eight-sided dice are 5d8, etc. Any additions to or subtractions from the die or dice are expressed after the identification, thus: d8 8 means a linear number grouping between 9 and 16, while 3d6 -2 means a bell-shaped progression from 1 to 16, with the greatest probability group in the middle (8, 9). This latter progression has the same median numbers as 2d6, but it has higher and loyver ends and a greater probability of a median number than if 2dl2 were used. When percentage dice are to be used; this is indicated by d%.
The 64 can be used to generate 25% incremental probabilities, random numbers from 1 to 4, with 1 it generates a linear 2-5, etc. It can be used to get 1 or 2 (1 or 2 = 1, 3 or 4 = 2) or in conjunction with any other dice to get linear or bell-shaped probability curves. For example, 2d4 = 2-8, 3d4 ^ 3-12, d4 d6 = 2-10, d4 d20 (as dlO) = 2-M.'when rolled in conjunction with another die, the d4 can be used to determine linear number ranges twice that shown on the other die, thus: d4 reading 1 or 2 means that whatever is read on the other die is the number shown; but if the d4 reads 3 or 4, add the highest number on the second die to the number shown — so if d8 is the second die 1 to 16 can be generated if a dl 2 is used 1 to 24 can be generated. If a d20 is used either 1 -20 (assuming the use of a standard d20 which is numbered 0-9 twice without coloring one set of faces to indicate that those faces have 10 added to the number appearing) or 1-40 (assuming that one set of faces is colored) can be gotten by adding 0 if 1 or 2 is rolled on the d4 and 10 or 20 (depending on die type) if a 3 or 4 is rolled. Linear series above this are possible simply bv varying the meaning of the d4 number; 1 always means add 0, but 2 can be interpreted as add the value (highest number) of the second die, 3 can be twice value, and 4 can be thrice value. Thus, a d4 reading 4 in conjunction with a d8 (linear curve 1-32) would mean 24 d8, or 25-32.
What applies to d4 has similar application with regard to d6, d8, dl2, and d20. The d6 has 16 2/3% intervals, d8 has 12 1/2% intervals, and d20 can have (1-2 = 1,3-4 = 2,5-6 = 3), while 1 to 5 can be easily read from a d20 (1-2 = 1, 3-4 = 2,5-6 = 3,7-8 = 4,9-0 = 5).
The d20 is used often, both as dlO and d20. The bell-shaped probability curves typically range from 2-20 to 5-50, i.e., 2, 3, 4 or 5d20 added together. Also common is the reading as above with one decimal place added to the result to get 20-200, 30-300, etc. In the latter case, a roll of 3 on one die and 0 (read as JO) totals 13, plus one place, or 130.
Non-platonic solid-shaped dice are available in some places. The most common of these is a ten-sided die numbered 0-9. As with the d20, this con be used for many purposes, even replacing the d20 if a second die is used in conjunction to get 5% interval curves (1-20). Also, the die can give 0-9 linear curve random numbers, as the d20 can.
Other dice available are various forms of "averaging" dice. The most common of these has six faces which read: 2, 3, 3, 4, 4, 5. The median of the curve it generates is still 3.5, that of a normal d6, but the low and high numbers, 2 and 5, are only half as likely to appear as 3 or 4. There is a 33 1/3% chance for either of the two latter numbers to be rolled, so the probabilities of absolutely average rolls are far greater. Other such dice have zeros on them, several low numbers, and so on. These sorts of dice, along with poker dice, "put & take" dice, or any other sort can be added in order to give you more flexibility or changing probabilities in random selection or event interpretation. For example:
The author has a d6 with the following faces: SPADE, CLUB, CLUB, DIAMOND, DIAMOND, HEART. If, during an encounter, players meet a character whose reaction is uncertain, the card suit die is rolled in conjunction with 3d6. Black suits mean dislike, with the SPADE equalling hate, while red equals like, the HEART being great favor. The 3d6 give a bell-shaped probability curve of 3-18, with 9-12 being the mean spread. SPADE 18 means absolute and unchangeable hate, while HEART 18 indicates the opposite. CLUBS or DIAMONDS can be altered by discourse, rewards, etc. | Thus, CLUBS 12 could possibly be altered to CLUBS 3 by offer of a tribute or favor, CLUBS 3 changed to DIAMONDS 3 by a gift, etc.
In closing this discussion, simply keep in mind that the dice are your tools. Learn to use them properly, and they will serve you well.
Bayes’s much more famous work, “An Essay toward Solving a Problem in the Doctrine of Chances,”24 was not published until after his death, when it was brought to the Royal Society’s attention in 1763 by a friend of his named Richard Price. It concerned how we formulate probabilistic beliefs about the world when we encounter new data.
Price, in framing Bayes’s essay, gives the example of a person who emerges into the world (perhaps he is Adam, or perhaps he came from Plato’s cave) and sees the sun rise for the first time. At first, he does not know whether this is typical or some sort of freak occurrence. However, each day that he survives and the sun rises again, his confidence increases that it is a permanent feature of nature. Gradually, through this purely statistical form of inference, the probability he assigns to his prediction that the sun will rise again tomorrow approaches (although never exactly reaches) 100 percent.
The official position of the USGS is even more emphatic: earthquakes cannot be predicted. “Neither the USGS nor Caltech nor any other scientists have ever predicted a major earthquake,” the organization’s Web site asserts.24 “They do not know how, and they do not expect to know how any time in the foreseeable future.”
Earthquakes cannot be predicted? This is a book about prediction, not a book that makes predictions, but I’m willing to stick my neck out: I predict that there will be more earthquakes in Japan next year than in New Jersey. And I predict that at some point in the next one hundred years, a major earthquake will hit somewhere in California.
Both the USGS and I are playing some semantic games. The terms “prediction” and “forecast” are employed differently in different fields; in some cases, they are interchangeable, but other disciplines differentiate them. No field is more sensitive to the distinction than seismology. If you’re speaking with a seismologist:
1. A prediction is a definitive and specific statement about when and where an earthquake will strike: a major earthquake will hit Kyoto, Japan, on June 28.
2. Whereas a forecast is a probabilistic statement, usually over a longer time scale: there is a 60 percent chance of an earthquake in Southern California over the next thirty years.
The USGS’s official position is that earthquakes cannot be predicted. They can, however, be forecasted.
Base 26 is one of two fairly natural ways of representing numbers as text using a 26-letter alphabet. The number of interest is expressed numerically in base 26, and then the 26 different base-26 digits are identified with letters as 0=A, 1=B, 2=C, ... 25=Z. Here are the first 100 digits of pi expressed in this way:
Lo! At the 6th digit we find a two-letter word (LO), and only a few digits later we find the three-letter ROD embedded in the four-letter TROD. How many other English words can be found if we continue looking?
First, a few π facts are in order. The digits of π (in any base) not only go on forever but behave statistically like a sequence of uniform random numbers. (Mathematically proving that this is the case - the "π is normal conjecture" - is a deep unsolved problem, but numerical analysis of several billion digits suggests that it is true.) Consequently, π in base 26 emulates the mythical army of typing monkeys spewing out random letters. Among other things, this implies that any text, no matter how long, should eventually appear in the base-26 digits of π!
We should expect to need about 2.5 x 1018 letters in order to find the phrase TO BE OR NOT TO BE (without the spaces) once. We can only get as far as TO BE in the first million.
That the first 6-letter word is OXYGEN suggests that π is truly the very stuff of life!
Though this does not seem to be a useful way of looking at all the digits of π, we mustn't fail to note one last logological property. Write π as usual in decimal, and group the digits as follows:
3. 14 15 9 26 5...
and then make the obvious substitution A=1, B=2, etc. You get C.NOIZE, which is rather fitting, because the random nature of π's digits means that when you look at it you SEE NOISE!
Life expectancy for a healthy American man of my age is about 90. (That’s not to be confused with American male life expectancy at birth, only about 78.) If I’m to achieve my statistical quota of 15 more years of life, that means about 15 times 365, or 5,475, more showers. But if I were so careless that my risk of slipping in the shower each time were as high as 1 in 1,000, I’d die or become crippled about five times before reaching my life expectancy. I have to reduce my risk of shower accidents to much, much less than 1 in 5,475.
This calculation illustrates the biggest single lesson that I’ve learned from 50 years of field work on the island of New Guinea: the importance of being attentive to hazards that carry a low risk each time but are encountered frequently.
Consider: If you’re a New Guinean living in the forest, and if you adopt the bad habit of sleeping under dead trees whose odds of falling on you that particular night are only 1 in 1,000, you’ll be dead within a few years. In fact, my wife was nearly killed by a falling tree last year, and I’ve survived numerous nearly fatal situations in New Guinea.
I now think of New Guineans’ hypervigilant attitude toward repeated low risks as “constructive paranoia”: a seeming paranoia that actually makes good sense. Now that I’ve adopted that attitude, it exasperates many of my American and European friends. But three of them who practice constructive paranoia themselves — a pilot of small planes, a river-raft guide and a London bobby who patrols the streets unarmed — learned the attitude, as I did, by witnessing the deaths of careless people.
Having learned both from those studies and from my New Guinea friends, I’ve become as constructively paranoid about showers, stepladders, staircases and wet or uneven sidewalks as my New Guinea friends are about dead trees. As I drive, I remain alert to my own possible mistakes (especially at night), and to what incautious other drivers might do.
My hypervigilance doesn’t paralyze me or limit my life: I don’t skip my daily shower, I keep driving, and I keep going back to New Guinea. I enjoy all those dangerous things. But I try to think constantly like a New Guinean, and to keep the risks of accidents far below 1 in 1,000 each time.
Her [Nightingale's] statistics were more than a study, they were indeed her religion. For her, Quetelet was the hero as scientist, and the presentation copy of his Physique Sociale is annotated by her on every page. Florence Nightingale believed—and in all the actions of her life acted upon that belief—that the administrator could only be successful if he were guided by statistical knowledge. The legislator—to say nothing of the politician—too often failed for want of this knowledge. Nay, she went further: she held that the universe—including human communities—was evolving in accordance with a divine plan; that it was man's business to endeavour to understand this plan and guide his actions in sympathy with it. But to understand God's thoughts, she held we must study statistics, for these are the measure of his purpose. Thus the study of statistics was for her a religious duty.
Any experiment may be regarded as forming an individual of a 'population' of experiments which might be performed under the same conditions. A series of experiments is a sample drawn from this population.
Now any series of experiments is only of value in so far as it enables us to form a judgment as to the statistical constants of the population to which the experiments belong. In a great number of cases the question finally turns on the value of a mean, either directly, or as the mean difference between the two qualities.
If the number of experiments be very large, we may have precise information as to the value of the mean, but if our sample be small, we have two sources of uncertainty:— (I) owing to the 'error of random sampling' the mean of our series of experiments deviates more or less widely from the mean of the population, and (2) the sample is not sufficiently large to determine what is the law of distribution of individuals.
The Charms of Statistics.—It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence. Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalised, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary. They are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man.
The determining cause of most wars in the past has been, and probably will be of all wars in the future, the uncertainty of the result; war is acknowledged to be a challenge to the Unknown, it is often spoken of as an appeal to the God of Battles. The province of science is to foretell; this is true of every department of science. And the time must come—how soon we do not know—when the real science of war, something quite different from the application of science to the means of war, will make it possible to foresee with certainty the issue of a projected war. That will mark the end of battles; for however strong the spirit of contention, no nation will spend its money in a fight in which it knows it must lose.
Statistics are far from being the barren array of figures ingeniously and laboriously combined into columns and tables, which many persons are apt to suppose them. They constitute rather the ledger of a nation, in which, like the merchant in his books, the citizen can read, at one view, all of the results of a year or of a period of years, as compared with other periods, and deduce the profit or the loss which has been made, in morals, education, wealth or power.